Weakly Regular Graph

Author

Eric W. Weisstein
October 30, 2017

This notebook downloaded from http://mathworld.wolfram.com/notebooks/GraphTheory/WeaklyRegularGraph.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/WeaklyRegularGraph.html.

©2017 Wolfram Research, Inc. except for portions noted otherwise

Counts

All

A076434

0, 0, 0, 0, 0, 2, 4, 16, 21

Total[Boole[Get[#]]]&/@FileNames["*.m",FileNameJoin[{$GraphDataDirectory,"Nodes","WeaklyRegular"}]]

Length/@Table[Select[Graphs[n],WeaklyRegularGraphQ],{n,8}]//Timing

Length/@Table[GraphData["WeaklyRegular",n],{n,10}]

TextGrid[Table[{n,Row[Riffle[GraphData[#,"Notation"]&/@GraphData["WeaklyRegular",n],","]]},{n,7}],Dividers->All]

GraphData["WeaklyRegular",7]

GraphData[{"Quartic",{7,1}},"StandardNames"]

GraphData[{"Quartic",{7,1}},"Classes"]

GraphData[{"Quartic",{7,1}},"Graph","All"]

Connected

A294405

Table[Total[Boole[Get[FileNameJoin[{$GraphDataDirectory,"Nodes","WeaklyRegular","WeaklyRegular_"<>If[n<10,"0",""]<>ToString[n]<>".m"}]][[ConnectedGraphIndices[n]]]]],{n,10}]

Table[Total[Boole[WeaklyRegularGraphQ/@ConnectedGraphs[n]]],{n,8}]//Timing

Table[Length@GraphData[{"WeaklyRegular","Connected"},n],{n,10}]

TextGrid[Table[{n,Row[Riffle[GraphData[#,"Notation"]&/@GraphData[{"Connected","WeaklyRegular"},n],","]]},{n,7}],Dividers->All]

Disconnected

Table[Total[Boole[Get[FileNameJoin[{$GraphDataDirectory,"Nodes","WeaklyRegular","WeaklyRegular_"<>If[n<10,"0",""]<>ToString[n]<>".m"}]][[DisconnectedGraphIndices[n]]]]],{n,10}]

Table[Total[Boole[WeaklyRegularGraphQ/@Graphs[n][[DisconnectedGraphIndices[n]]]]],{n,8}]//Timing

Table[Length@GraphData[{"WeaklyRegular","Disconnected"},n],{n,10}]

Table

Big Table

wr=Sort[{"WeaklyRegular"/.GraphData[#,"NotationRules"],#}&/@GraphData["WeaklyRegular"]]

TableForm[{First[#],GraphData[Last[#],"Name"]}&/@DeleteCases[Cases[wr,{_,_String}],{_,_?(StringMatchQ[#,RegularExpression["(CrossingNumber|Two|Three|Four|Five|Foster).*"]]&)}],TableDepth->2]//TraditionalForm

Graphic

V5

ShowGraphArray[Flatten[Partition[#,8,8,{1,1},{}]&/@DeleteCases[sr=Map[RecognizeGraph,Table[Select[Graphs[n],RegularButNotStronglyRegularQ],{n,8}],{2}],
{}],1],
VertexStyle->Disk[.1],
VertexColor->Red]//Timing

V7

wr=#[[All,2]]&/@SplitBy[Sort[GraphData[#,{"V","#"}]&/@GraphData["WeaklyRegular",;;8]],First]

Length/@wr

(wr8=Monitor[Select[GetGraphData[8],MemberQ[GetGraphData[g=#,"Classes"],"WeaklyRegular"]&],g])//Timing

Length[wr8]

wr678=Append[Take[wr,2],wr8]

GraphicsGrid[Map[GraphPlot[GetGraph[#],Method->None,PlotLabel->TextCell[GetGraphData[#,"Name"],PageWidth->80,TextAlignment->Center],BaseStyle->{Italic,9}]&,Flatten[Partition[#,5,5,{1,1},{}]&/@wr,1],{2}],ImageSize->400,Spacings->{-40,20}]

Checks

DeleteCases[Monitor[{g=#,WeaklyRegularGraphQ[GraphData[g]],GraphData[#,"WeaklyRegular"]}&/@GraphData[;;100],g],{_,x_,x_}]//Timing